I'm going to take a stab at this! Remember, this is trying to explain to a kid without using complicated maths or formal terminalogy.
First you need to explain how the speed of points on a spinning object changes with distance from the axis of rotation. Imagine (or better, draw!) looking down on the spinning top from above, when it is spinning perfectly balanced. It turns at a certain speed, a number of turns per second. Now imagine a dot on the outer edge of the spinning top, and another dot halfway toward the center. The two dots must do the same number of turns per second around the middle, but the outer dot travels in a bigger circle. It has further to go, so it must go faster in order to get all the way around in the same time.
Next, to explain how two different rotations create a third rotation. This is probably the hardest part to explain. Imagine a wheel mounted on a pole, like this diagram from wikipedia:
The wheel is mounted vertically and can rotate around a horizontal axis, in addition the pole can spin the wheel around a vertical axis. Now imagine the pole is spinning the whole wheel around the vertical axis, similar to a spinning top. Imagine a point on the wheel near the outer edge (dm1 on this diagram) and imagine you rotate the wheel to bring this point toward the top. The way the wheel is spinning means the the circle the dot makes as it gets nearer the top must get smaller, so that part of the wheel has to slow down. It's going too fast for the smaller circle, so it creates a 'pull' on the wheel, shown by the arrow. Now imagine a point at the top of the wheel (dm2) and you rotate the wheel to bring it toward the outer edge. The circle made by this part of the wheel is getting bigger, so this part of the wheel has to speed up. It's going too slow for the bigger circle, so it creates a 'drag' on the wheel, shown by the second arrow. The way the wheel is being spun by the pole means that the left half and the right half of the wheel are moving in opposite directions, so a pull on one side and a drag on the other are in the same direction.
Now you can show the same thing for points on the bottom half of the wheel, except that the pull and the drag on the bottom half are in the opposite direction to those on the top half. Pulling in different directions on the top and bottom of the wheel, makes the whole wheel want to flip over. Now you have shown that spinning something in one direction and then rotating it in a second direction makes it want to flip over in a third direction, at a right angle to the other two.
Finally you can use this to explain why the spinning top can't fall over. The spinning top is spinning about its vertical axis, like the wheel being spun around by the pole. If the spinning top isn't perfectly balanced, if it leans slightly in one direction, gravity will try to pull it over in that direction. This rotates the top in that direction. Just like when the wheel is rotated while the pole is spinning, this creates pulls and drags on parts of the spinning top that make it want to flip over in the third direction, at 90 degrees. So when gravity makes the top lean over one way, the flipping effect tries to make it lean over in a different direction. The spinning top is pulled in both directions at once so it will actually lean over in a direction half way between the two (draw a diagram if it helps!).
BUT! Here's the clever part! As soon as the top starts leaning in a different direction, that becomes the new direction that gravity tries to tip it over in. When the direction that gravity is trying to rotate the top in changes, the direction it tries to flip in also changes, to stay 90-degrees away. So the top is pulled in two new directions, and the direction it leans changes again. And the same thing happens, again and again... The result is that the direction in which the top leans constantly changes - it is pulled around in a circle! Every time gravity tries to pull the top over one way, the flipping effect makes it lean a different way. As long as the top keeps spinning, the flipping effect means it can't fall over!